Monadically stable classes of graphs not only subsume nowhere dense
classes, but also include every class that interprets (by a simple
interpretation) in a nowhere dense class. This implies that
monadically stable classes include also dense classes of graphs, such
as complements and fixed powers of nowhere dense classes. In fact, it
has been conjectured that every monadically stable class is a simple
interpretation of a nowhere dense
class [1]}. Monadically NIP classes further
generalize monadically stable classes and most prominently also
include classes of bounded clique- and twinwidth
[2]}, [3]}. These outlined
connections with model theory have recently triggered the interest to
generalize combinatorial and algorithmic results from nowhere dense
classes to simple interpretations of nowhere dense, monadically stable
and monadically NIP classes, and ultimately to solve the
model-checking problem for first-order logic on these
classes [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [1]}, [11]}, [12]}. Results on monadically stable and monadically
NIP classes in classical model theory
include [13]}, [14]}, [15]}, [16]}. More background on general NIP theories can be
found in the textbook [17]}.
